3.4: Rational Numbers (2024)

Tech Check: Using Desmos to Find the Square Root of a Number

When Desmos is used, there is a tab at the bottom of the screen that opens the keyboard for Desmos. The keyboard is shown below. On the keyboard (Figure 3.24) is the square root symbol $\left(\sqrt{}\right)$. To find the square root of a number, click the square root key, and then type the number. Desmos will automatically display the value of the square root as you enter the number.

Example 3.51: Identifying Perfect Squares

Which of the following are perfect squares?

1. 45
2. 144

Answer

1. We could attempt to find the perfect square by factoring. Writing all the factor pairs of 45 results in $1\times 45,3\times 15$, and $5\times 9$. None of the pairs is a square, so 45 is not a perfect square. Using a calculator to find the square root of 45, we obtain 6.708 (rounded to three decimal places). Since this was not an integer, the original number was not a perfect square.
2. We could attempt to find the perfect square by factoring. Writing all the factor pairs of 144 results in $1\times 144,2\times 72,3\times 48,6\times 24,8\times 18$, and $12\times 12$. Since the last pair is an integer multiplied by itself, 144 is a perfect square. Alternately, using Desmos to find the square root of 144, we obtain 12. Since the square root of 144 is an integer, 144 is a perfect square.

Your Turn 3.51

Determine if the following are perfect squares:

1. 45
2. 144

Video

Introduction to Fractions

Example 3.52: Identifying Rational Numbers

Determine which of the following are rational numbers:

1. $\sqrt{73}$
2. $4.556$
3. $3\frac{1}{5}$
4. $\frac{41}{17}$
5. $5.\overline{64}$

Answer

1. Since 73 is not a perfect square, its square root is not a rational number. This can also be seen when a calculator is used. Entering $\sqrt{73}$ into a calculator results in 8.544003745317 (and then more decimal values after that). There is no repeated pattern, so this is not a rational number.
2. Since 4.556 is a decimal that terminates, this is a rational number.
3. $3\frac{1}{5}$ is a mixed number, so it is a rational number.
4. $\frac{41}{17}$ is an integer divided by an integer, so it is a rational number.
5. $\mathrm{5.646464...}$ is a decimal that repeats a pattern, so it is a rational number.

Your Turn 3.52

Determine which of the following are rational numbers:

\(\sqrt{13}\)

\(-13 . \overline{2} \overline{1}\)

\(\frac{-48}{-16}\)

\(-4 \frac{18}{19}\)

14.1131

Example 3.53: Determining If Two Fractions Are Equivalent

Determine if $\frac{12}{30}$ and $\frac{14}{35}$ are equivalent fractions.

Answer

Applying the definition, $a=12,b=30,c=$ and $d=$. So $a\times d=12\times 35=$. Also, $b\times c=30\times 14=$. Since these values are equal, the fractions are equivalent.

Your Turn 3.53

Determine if \(\frac{8}{14}\) and \(\frac{12}{26}\) are equivalent fractions.

Video

Equivalent Fractions

Example 3.54: Reducing Fractions to Lowest Terms

Express the following rational numbers in lowest terms:

1. $\frac{36}{48}$
2. $\frac{100}{250}$
3. $\frac{51}{136}$

Answer

1. One process to reduce $\frac{36}{48}$ to lowest terms is to identify the GCD of 36 and 48 and divide out the GCD. The GCD of 36 and 48 is 12.

   Step 1: We can then rewrite the numerator and denominator by factoring 12 from both.

   $\frac{36}{48}=\frac{12\times 3}{12\times 4}$

   Step 2: We can now divide out the 12s from the numerator and denominator.

   $\frac{36}{48}=\frac{\cancel{12}\times 3}{\cancel{12}\times 4}=\frac{3}{4}$

   So, when $\frac{36}{48}$ is reduced to lowest terms, the result is $\frac{3}{4}$.

   Alternately, you could identify a common factor, divide out that common factor, and repeat the process until the remaining fraction is in lowest terms.

   Step 1: You may notice that 4 is a common factor of 36 and 48.

   Step 2: Divide out the 4, as in $\frac{36}{48}=\frac{4\times 9}{4\times 12}=\frac{\cancel{4}\times 9}{\cancel{4}\times 12}=\frac{9}{12}$.

   Step 3: Examining the 9 and 12, you identify 3 as a common factor and divide out the 3, as in $\frac{9}{12}=\frac{\cancel{3}\times 3}{\cancel{3}\times 4}=\frac{3}{4}$. The 3 and 4 have no common positive factors other than 1, so it is in lowest terms.

   So, when $\frac{36}{48}$ is reduced to lowest terms, the result is $\frac{3}{4}$.

2. Step 1: To reduce $\frac{100}{250}$ to lowest terms, identify the GCD of 100 and 250. This GCD is 50.

   Step 2: We can then rewrite the numerator and denominator by factoring 50 from both.

   $\frac{100}{250}=\frac{50\times 2}{50\times 5}$.

   Step 3: We can now divide out the 50s from the numerator and denominator.

   $\frac{100}{250}=\frac{\cancel{50}\times 2}{\cancel{50}\times 5}=\frac{2}{5}$

   So, when $\frac{100}{250}$ is reduced to lowest terms, the result is $\frac{2}{5}$.

3. Step 1: To reduce $\frac{51}{136}$ to lowest terms, identify the GCD of 51 and 136. This GCD is 17.

   Step 2: We can then rewrite the numerator and denominator by factoring 17 from both.

   $\frac{51}{136}=\frac{17\times 3}{17\times 8}$

   Step 3: We can now divide out the 17s from the numerator and denominator.

   $\frac{51}{136}=\frac{\cancel{17}\times 3}{\cancel{17}\times 8}=\frac{3}{8}$

   So, when $\frac{51}{136}$ is reduced to lowest terms, the result is $\frac{3}{8}$.

Your Turn 3.54

Express \(\frac{252}{840}\) in lowest terms.

Video

Reducing Fractions to Lowest Terms

Tech Check: Using Desmos to Find Lowest Terms

Desmos is a free online calculator. Desmos supports reducing fractions to lowest terms. When a fraction is entered, Desmos immediately calculates the decimal representation of the fraction. However, to the left of the fraction, there is a button that, when clicked, shows the fraction in reduced form.

Video

Using Desmos to Reduce a Fraction

Tech Check: Using Desmos to Add Rational Numbers in Fractional Form

To create a fraction in Desmos, enter the numerator, then use the division key (/) on your keyboard, and then enter the denominator. The fraction is then entered. Then click the right arrow key to exit the denominator of the fraction. Next, enter the arithmetic operation (+ or –). Then enter the next fraction. The answer is displayed dynamically (calculates as you enter). To change the Desmos result from decimal form to fractional form, use the fraction button (Figure 3.26) on the left of the line that contains the calculation:

Example 3.55: Adding Rational Numbers Using Desmos

Calculate $\frac{23}{42}+\frac{9}{56}$ using Desmos.

Answer

Enter $\frac{23}{42}+\frac{9}{56}$ in Desmos. The result is displayed as $0.70833333333$ (which is $0.708\overline{3}$). Clicking the fraction button to the left on the calculation line yields $\frac{17}{24}$.

Your Turn 3.55

Calculate \(\frac{124}{297}+\frac{3}{125}\).

FORMULA

If $c$ is a non-zero integer, then $\frac{a}{c}\pm \frac{b}{c}=\frac{a\pm }{}$.

Example 3.56: Adding Rational Numbers with the Same Denominator

Calculate $\frac{13}{28}+\frac{7}{28}$.

Answer

Since the rational numbers have the same denominator, we perform the addition of the numerators, $13+7$, and then place the result in the numerator and the common denominator, 28, in the denominator. $\frac{13}{28}+\frac{7}{28}=\frac{13+7}{28}=\frac{20}{28}$

Once we have that result, reduce to lowest terms, which gives $\frac{20}{28}=\frac{4\times 5}{4\times 7}=\frac{\cancel{4}\times 5}{\cancel{4}\times 7}=\frac{5}{7}$.

Your Turn 3.56

Calculate \(\frac{38}{73}+\frac{7}{73}\).

Example 3.57: Subtracting Rational Numbers with the Same Denominator

Calculate $\frac{45}{136}-\frac{17}{136}$.

Answer

Since the rational numbers have the same denominator, we perform the subtraction of the numerators, $45-17$, and then place the result in the numerator and the common denominator, 136, in the denominator. $\frac{45}{136}-\frac{17}{136}-\frac{45-17}{136}=\frac{28}{136}$

Once we have that result, reduce to lowest terms, this gives $\frac{28}{136}=\frac{4\times 7}{4\times 34}=\frac{\cancel{4}\times 7}{\cancel{4}\times 34}=\frac{7}{34}$.

Your Turn 3.57

1. Calculate \(\frac{21}{40}-\frac{8}{40}\).

Example 3.58: Adding Rational Numbers with Unequal Denominators

Calculate $\frac{11}{18}+\frac{2}{15}$.

Answer

The denominators of the fractions are 18 and 15, so we label $b=$ and $d=$.

Step 1: Find LCM(18,15). This is 90.

Step 2: Calculate $n$ and $m$. $n=\frac{90}{18}=5$ and $m=\frac{90}{15}=6$.

Step 3: Multiplying the numerator and denominator of $\frac{11}{18}$ by $n=5$ yields $\frac{11\times 5}{18\times 5}=\frac{55}{90}$.

Step 4: Multiply the numerator and denominator of $\frac{2}{15}$ by $m=6$ yields $\frac{2\times 6}{15\times 6}=\frac{12}{90}$.

Step 5: Now we add the values from Steps 3 and 4: $\frac{55}{90}+\frac{12}{90}=\frac{67}{90}$.

This is in lowest terms, so we have found that $\frac{11}{18}+\frac{2}{15}=\frac{67}{90}$.

Your Turn 3.58

Calculate \(\frac{4}{9}+\frac{7}{12}\).

Example 3.59: Subtracting Rational Numbers with Unequal Denominators

Calculate $\frac{14}{25}-\frac{9}{70}$.

Answer

The denominators of the fractions are 25 and 70, so we label $b=$ and $d=$.

Step 1: Find LCM(25,70). This is 350.

Step 2: Calculate $n$ and $m$: $n=\frac{350}{25}=$ and $m=\frac{350}{70}=5$.

Step 3: Multiplying the numerator and denominator of $\frac{14}{25}$ by $n=$ yields $\frac{14\times 14}{25\times 14}=\frac{196}{350}$.

Step 4: Multiplying the numerator and denominator of $\frac{9}{70}$ by $m=5$ yields $\frac{9\times 5}{70\times 5}=\frac{45}{350}$.

Step 5: Now we subtract the value from Step 4 from the value in Step 3: $\frac{196}{350}-\frac{45}{350}=\frac{151}{350}$.

This is in lowest terms, so we have found that $\frac{14}{25}-\frac{9}{70}=\frac{151}{350}$.

Your Turn 3.59

Calculate \(\frac{10}{99}-\frac{17}{300}\).

Video

Adding and Subtracting Fractions with Different Denominators

Example 3.60: Rewriting an Improper Fraction as a Mixed Number

Rewrite $\frac{48}{13}$ as a mixed number.

Answer

When 48 is divided by 13, the result is 3 with a remainder of 9. So, we can rewrite $\frac{48}{13}$ as $3\frac{9}{13}$.

Your Turn 3.60

Rewrite \(\frac{95}{26}\) as a mixed number.

Video

Converting an Improper Fraction to a Mixed Number Using Desmos

Example 3.61: Rewriting a Mixed Number as an Improper Fraction

Rewrite $5\frac{4}{9}$ as an improper fraction.

Answer

Step 1: Multiply the integer part, 5, by the denominator, 9, which gives $5\times 9=45$.

Step 2: Add that product to the numerator, which gives $45+4=49$.

Step 3: Write the number as the sum, 49, divided by the denominator, 9, which gives $\frac{49}{9}$.

Your Turn 3.61

Rewrite \(9 \frac{5}{14}\) as an improper fraction.

Tech Check: Using Desmos to Rewrite a Mixed Number as an Improper Fraction

Desmos can be used to convert from a mixed number to an improper fraction. To do so, we use the idea that a mixed number, such as $5\frac{6}{11}$, is another way to represent $5+\frac{6}{11}$. If $5+\frac{6}{11}$ is entered in Desmos, the result is the decimal form of the number. However, clicking the fraction button to the left will convert the decimal to an improper fraction, $\frac{61}{11}$. As an added bonus, Desmos will automatically reduce the fraction to lowest terms.

Example 3.62: Rounding Off a Number in Decimal Form to Three Digits

Round 5.67849 to three decimal digits.

Answer

The third decimal digit is 8. The digit following the 8 is 4. When the digit is 4, we write the number only to the third digit. So, 5.67849 rounded off to three decimal places is 5.678.

Your Turn 3.62

Round 5.1082 to three decimal places.

Rounding Off a Number in Decimal Form to Four Digits

Round 45.11475 to four decimal digits.

Answer

The fourth decimal digit is 7. The digit following the 7 is 5. When the digit is 5, we write the number only to the fourth decimal digit, 45.1147. We then replace the fourth decimal digit by one more than the fourth digit, which yields 45.1148. So, 45.11475 rounded off to four decimal places is 45.1148.

Your Turn 3.63

Round 18.6298 to two decimal places.

Example 3.64: Converting a Rational Number in Fraction Form into Decimal Form

Convert $\frac{47}{25}$ into decimal form.

Answer

Using a calculator to divide 47 by 25, the result is 1.88.

Your Turn 3.64

Convert \(\frac{48}{30}\) into decimal form.

Example 3.65: Converting from Decimal Form to Fraction Form with Terminating Decimals

Convert 3.2117 to fraction form.

Answer

Step 1: There are four digits after the decimal point, so $n=4$.

Step 2: Raise 10 to the fourth power, ${10}^4=\mathrm{10,000}$.

Step 3: When we remove the decimal point, we have 32,117.

Step 4: The fraction has as its numerator the result from Step 3 and as its denominator the result of Step 2, which is the fraction $\frac{\mathrm{32,117}}{\mathrm{10,000}}$.

Your Turn 3.65

Convert 17.03347 to fraction form.

FORMULA

If $b$ and $d$ are non-zero integers, then $\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}$.

Example 3.66: Multiplying Rational Numbers

Calculate $\frac{12}{25}\times \frac{10}{21}$.

Answer

Multiply the numerators and place that in the numerator, and then multiply the denominators and place that in the denominator.

$\frac{12}{25}\times \frac{10}{21}=\frac{12\times 10}{25\times 21}=\frac{120}{525}$

This is not in lowest terms, so this needs to be reduced. The GCD of 120 and 525 is 15.

$\frac{120}{525}=\frac{15\times 8}{15\times 35}=\frac{8}{35}$

Your Turn 3.66

Calculate \(\frac{45}{88} \times \frac{28}{75}\).

Video

Multiplying Fractions

Example 3.67: Dividing Decimals with a Calculator

Calculate 3.45 ÷ 2.341 using a calculator. Round to three decimal places if necessary.

Answer

Using a calculator, we obtain 1.473729175565997. Rounding to three decimal places we have 1.474.

Your Turn 3.67

Calculate \(45.63 \div 17.13\) using a calculator. Round to three decimal places, if necessary.

FORMULA

If $b,c$ and $d$ are non-zero integers, then $\frac{b}{a}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=$.

Example 3.68: Dividing Rational Numbers

1. Calculate $\frac{4}{21}÷\frac{6}{35}$.
2. Calculate $\frac{1}{8}÷\frac{5}{28}$.

Answer

1. Step 1: Find the reciprocal of the number being divided by $\frac{6}{35}$. The reciprocal of that is $\frac{35}{6}$.

   Step 2: Multiply the first fraction by that reciprocal.

   $\frac{4}{21}÷\frac{6}{35}=\frac{4}{21}\times \frac{35}{6}=\frac{140}{126}$

   The answer, $\frac{140}{126}$ is not in lowest terms. The GCD of 140 and 126 is 14. Factoring and canceling gives $\frac{140}{126}=\frac{14\times 10}{14\times 9}=\frac{10}{9}$.

2. Step 1: Find the reciprocal of the number being divided by, which is $\frac{5}{28}$. The reciprocal of that is $\frac{28}{5}$.

   Step 2: Multiply the first fraction by that reciprocal: $\frac{1}{8}÷\frac{5}{28}=\frac{1}{8}\times \frac{28}{5}=\frac{28}{40}$

   The answer, $\frac{28}{40}$, is not in lowest reduced form. The GCD of 28 and 40 is 4. Factoring and canceling gives $\frac{28}{40}=\frac{4\times 7}{4\times 10}=\frac{7}{10}$.

Your Turn 3.68

1. Calculate \(\frac{46}{175} \div \frac{69}{285}\).

Video

Dividing Fractions

Example 3.69: Applying the Order of Operations with Rational Numbers

Correctly apply the rules for the order of operations to accurately compute $(\frac{5}{7}-\frac{2}{7})\times 2^3$.

Answer

Step 1: To calculate this, perform all calculations within the parentheses before other operations.

$(\frac{5}{7}-\frac{2}{7})\times 2^3=\left(\frac{3}{7}\right)\times 2^3$

Step 2: Since all parentheses have been cleared, we move left to right, and compute all the exponents next.

$\left(\frac{3}{7}\right)\times 2^3=\left(\frac{3}{7}\right)\times 8$

Step 3: Now, perform all multiplications and divisions, moving left to right.

$\left(\frac{3}{7}\right)\times 8=\frac{24}{7}$

Your Turn 3.69

\(\begin{array}{l}
\text { Correctly apply the rules for the order of operations to accurately compute }\\
\left(\frac{3}{16}+\frac{7}{16}\right)^2+\frac{1}{5} \div \frac{3}{10}
\end{array}\)

Example 3.70: Applying the Order of Operations with Rational Numbers

Correctly apply the rules for the order of operations to accurately compute $4+\frac{2}{3}÷{\left({\left(\frac{5}{9}\right)}^2-\left(\frac{2}{3}+5\right)\right)}^2$.

Answer

To calculate this, perform all calculations within the parentheses before other operations. Evaluate the innermost parentheses first. We can work separate parentheses expressions at the same time.

Step 1: The innermost parentheses contain $\frac{2}{3}+5$. Calculate that first, dividing after finding the common denominator.

$\begin{array}{l}4+\frac{2}{3}÷{\left({\left(\frac{5}{9}\right)}^2-\left(\frac{2}{3}+5\right)\right)}^2\\ =4+\frac{2}{3}÷{\left({\left(\frac{5}{9}\right)}^2-\left(\frac{2}{3}+\frac{5}{1}\right)\right)}^2\\ =4+\frac{2}{3}÷{\left({\left(\frac{5}{9}\right)}^2-\left(\frac{2}{3}+\frac{15}{3}\right)\right)}^2\\ =4+\frac{2}{3}÷{\left({\left(\frac{5}{9}\right)}^2-\left(\frac{17}{3}\right)\right)}^2\end{array}$

Step 2: Calculate the exponent in the parentheses, ${\left(\frac{5}{9}\right)}^2$.

$\begin{array}{l}4+\frac{2}{3}÷{\left({\left(\frac{5}{9}\right)}^2-\left(\frac{17}{3}\right)\right)}^2\hfill \\ =4+\frac{2}{3}÷{\left(\left(\frac{25}{81}\right)-\left(\frac{17}{3}\right)\right)}^2\hfill \end{array}$

Step 3: Subtract inside the parentheses is done, using a common denominator.

$\begin{array}{l}4+\frac{2}{3}÷{\left(\left(\frac{25}{81}\right)-\left(\frac{17}{3}\right)\right)}^2\\ 4+\frac{2}{3}÷{\left(\left(\frac{25}{81}\right)-\left(\frac{17\times 27}{3\times 27}\right)\right)}^2\\ 4+\frac{2}{3}÷{\left(\left(\frac{25}{81}\right)-\left(\frac{459}{81}\right)\right)}^2\\ 4+\frac{2}{3}÷{\left(\left(\frac{-434}{81}\right)\right)}^2\end{array}$

Step 4: At this point, evaluate the exponent and divide.

$\begin{array}{l}4+\frac{2}{3}÷{\left(\left(\frac{-434}{81}\right)\right)}^2\\ 4+\frac{2}{3}÷\left(\frac{\mathrm{188,356}}{\mathrm{6,561}}\right)\\ =4+\frac{2}{3}\times \left(\frac{\mathrm{6,561}}{\mathrm{188,356}}\right)\\ =4+\frac{\mathrm{2,187}}{\mathrm{94,178}}\end{array}$

Step 5: Add.

$\begin{array}{l}4+\frac{\mathrm{2,187}}{\mathrm{94,178}}\\ =\frac{\mathrm{378,899}}{\mathrm{94,178}}\end{array}$

Had this been done on a calculator, the decimal form of the answer would be 4.0232 (rounded to four decimal places).

Your Turn 3.70

Correctly apply the rules for the order of operations to accurately compute

\(\left(\frac{3}{5}+2\right) \times\left(\frac{4}{5}-\frac{1}{2}\right)^2 \div \frac{11}{15}\).

Video

Order of Operations Using Fractions

Example 3.71: Applying the Density Property of Rational Numbers

Demonstrate the density property of rational numbers by finding a rational number between $\frac{4}{11}$ and $\frac{7}{12}$.

Answer

To find a rational number between $\frac{4}{11}$ and $\frac{7}{12}$:

Step 1: Add the fractions.

$\frac{4}{11}+\frac{7}{12}=\frac{4\times 12}{11\times 12}+\frac{7\times 11}{12\times 11}=\frac{48}{132}+\frac{77}{132}=\frac{125}{132}$

Step 2: Divide the result by 2. Recall that to divide by 2, you multiply by the reciprocal of 2. The reciprocal of 2 is $\frac{1}{2}$, as seen below.

$\frac{125}{132}÷2=\frac{125}{132}\times \frac{1}{2}=\frac{125}{264}$

So, one rational number between $\frac{4}{11}$ and $\frac{7}{12}$ is $\frac{125}{264}$.

We could check that the number we found is between the other two by finding the decimal representation of the numbers. Using a calculator, the decimal representations of the rational numbers are 0.363636…, 0.473484848…, and 0.5833333…. Here it is clear that $\frac{125}{264}$ is between $\frac{4}{11}$ and $\frac{7}{12}$.

Your Turn 3.71

Demonstrate the density property of rational numbers by finding a rational number between \(\frac{27}{13}\) and \(\frac{21}{10}\).

Example 3.72: Mixing Soil for Vegetables

James is mixing soil for a raised garden, in which he plans to grow a variety of vegetables. For the soil to be suitable, he determines that $\frac{2}{5}$ of the soil can be topsoil, but $\frac{2}{5}$ needs to be peat moss and $\frac{1}{5}$ has to be compost. To fill the raised garden bed with 60 cubic feet of soil, how much of each component does James need to use?

Answer

In this example, we know the proportion of each component to mix, and we know the total amount of the mix we need. In this kind of situation, we need to determine the appropriate amount of each component to include in the mixture. For each component of the mixture, multiply 60 cubic feet, which is the total volume of the mixture we want, by the fraction required of the component.

Step 1: The required fraction of topsoil is $\frac{2}{5}$, so James needs $60\times \frac{2}{5}$ cubic feet of topsoil. Performing the multiplication, James needs $60\times \frac{2}{5}=\frac{120}{5}=24$ (found by treating the fraction as division, and 120 divided by 5 is 24) cubic feet of topsoil.

Step 2: The required fraction of peat moss is also $\frac{2}{5}$, so he also needs $60\times \frac{2}{5}$ cubic feet, or $60\times \frac{2}{5}=\frac{120}{5}=24$ cubic feet of peat moss.

Step 3: The required fraction of compost is $\frac{1}{5}$. For the compost, he needs $60\times \frac{1}{5}=\frac{60}{5}=12$ cubic feet.

Your Turn 3.72

Ashley wants to study for 10 hours over the weekend. She plans to spend half the time studying math, a quarter of the time studying history, an eighth of the time studying writing, and the remaining eighth of the time studying physics. How much time will Ashley spend on each of those subjects?

Example 3.73: Determining the Number of Specialty Pizzas

At Bella’s Pizza, one-third of the pizzas that are ordered are one of their specialty varieties. If there are 273 pizzas ordered, how many were specialty pizzas?

Answer

One-third of the whole are specialty pizzas, so we need one-third of 273, which gives $\frac{1}{3}\times 273=\frac{273}{3}=91$, found by dividing 273 by 3. So, 91 of the pizzas that were ordered were specialty pizzas.

Your Turn 3.73

Danny, a nutritionist, is designing a diet for her client, Callum. Danny determines that Callum’s diet should be 30% protein. If Callum consumes 2,400 calories per day, how many calories of protein should Danny tell Callum to consume?

Video

Finding a Fraction of a Total

Example 3.74: Converting Liters to Gallons

It is known that 1 liter (L) is 0.264172 gallons (gal). Use this to convert 14 liters into gallons.

Answer

We know that 1 liter = 0.264172 gal. Since we are converting from liters, when we create the fraction we use, make sure the liter part of the equivalence is in the denominator. So, to convert the 14 liters to gallons, we multiply 14 by $\frac{1\mathrm{gal}}{0.264172\mathrm{gal}/1\mathrm{liter}}$. Notice the gallon part is in the numerator since we’re converting to gallons, and the liter part is in the denominator since we are converting from liters. Performing this and rounding to three decimal places, we find that 14 liters is $14\mathrm{liter}\times \frac{0.264172\mathrm{gal}}{1\mathrm{liter}}=3.69841\mathrm{gal}$.

Your Turn 3.74

One mile is equal to 1.60934 km. Convert 200 miles to kilometers. Round off the answer to three decimal places.

Example 3.75: Converting Centimeters to Inches

It is known that 1 inch is 2.54 centimeters. Use this to convert 100 centimeters into inches.

Answer

We know that 1 inch = 2.54 cm. Since we are converting from centimeters, when we create the fraction we use, make sure the centimeter part of the equivalence is in the denominator, $\frac{1\text{in}}{2.54\text{cm}}$. To convert the 100 cm to inches, multiply 100 by $\frac{1\text{in}}{2.54\text{cm}}$. Notice the inch part is in the numerator since we’re converting to inches, and the centimeter part is in the denominator since we are converting from centimeters. Performing this and rounding to three decimal places, we obtain $100\text{cm}\times \frac{1\text{in}}{2.54\text{cm}}=39.370\text{in}$. This means 100 cm equals 39.370 in.

Your Turn 3.75

It is known that 4 quarts equals 3.785 liters. If you have 25 quarts, how many liters do you have? Round off to three decimal places.

Video

Converting Units

Example 3.76: Rewriting a Percentage as a Fraction

Rewrite the following as fractions:

1. 31%
2. 93%

Answer

1. Using the definition and $n=$, 31% in fraction form is $\frac{31}{100}$.
2. Using the definition and $n=$, 93% in fraction form is $\frac{93}{100}$.

Your Turn 3.76

Rewrite the following as fractions:

1. 4%
2. 50%

Example 3.77: Rewriting a Percentage as a Decimal

Rewrite the following percentages in decimal form:

1. 54%
2. 83%

Answer

1. Using the definition and $n=$, 54% in fraction form is $\frac{54}{100}$. Dividing a number by 100 moves the decimal two places to the left; 54% in decimal form is then 0.54.
2. Using the definition and $n=$, 83% in fraction form is $\frac{83}{100}$. Dividing a number by 100 moves the decimal two places to the left; 83% in decimal form is then 0.83.

Your Turn 3.77

Rewrite the following percentages in decimal form:

1. 14%
2. 7%

FORMULA

$n\%$ of \(x\) items is $\frac{n}{100}\times x$. The \(x\) is referred to as the total, the $n$ is referred to as the percent or percentage, and the value obtained from $\frac{n}{100}\times x$ is the part of the total and is also referred to as the percentage of the total.

Example 3.78: Finding a Percentage of a Total

1. Determine 40% of 300.
2. Determine 64% of 190.

Answer

1. The total is 300, and the percentage is 40. Using the decimal form of 40% and multiplying we obtain $0.40\times 300=120$.
2. The total is 190, and the percentage is 64. Using the decimal form of 64% and multiplying we obtain $0.64\times 190=121.6$.

Your Turn 3.78

1. Determine 25% of 1,200.

2. Determine 53% of 1,588.

FORMULA

If we know that $n$% of the total is \(x\), then the total is $\frac{100\times x}{n}$.

Example 3.79: Finding the Total When the Percentage and Percentage of the Total Are Known

1. What is the total if 28% of the total is 140?
2. What is the total if 6% of the total is 91?

Answer

1. 28 is the percentage, so $n=$. 28% of the total is 140, so $x=$. Using those we find that the total was $\frac{100\times \text{140}}{\text{28}}=500$.
2. 6 is the percentage, so $n=6$. 6% of the total is 91, so $x=$. Using those we find that the total was $\frac{100\times \text{91}}{\text{6}}=\mathrm{1,516.6}$.

Your Turn 3.79

What is the total if 25% of the total is 30?

What is the total if 45% of the total is 360?

FORMULA

The percentage, $n$, of $b$ that is $a$ is $\frac{a}{b}\times 100\%$.

Example 3.80: Finding the Percentage When the Total and Percentage of the Total Are Known

Find the percentage in the following:

1. Total is 300, percentage of the total is 60.
2. Total is 440, percentage of the total is 176.

Answer

1. The total is 300; the percentage of the total is 60. Calculating yields 0.2. Moving the decimal two places to the right gives 20. Appending the percentage to this number results in 20%. So, 60 is 20% of 300.
2. The total is 440; the percentage of the total is 176. Calculating yields 0.4. Moving the decimal two places to the right gives 40. Appending the percentage to this number results in 40%. So, 176 is 40% of 440.

Your Turn 3.80

Find the percentage in the following:

Total is 1,000, percentage of the total is 70.

Total is 500, percentage of the total is 425.

Example 3.81: Percentage of Students Who Are Sleep Deprived

A study revealed that 70% of students suffer from sleep deprivation, defined to be sleeping less than 8 hours per night. If the survey had 400 participants, how many of those participants had less than 8 hours of sleep per night?

Answer

The percentage of interest is 70%. The total number of students is 400. With that, we can find how many were in the percentage of the total, or, how many were sleep deprived. Applying the formula from above, the number who were sleep deprived was $0.70\times 400=280$; 280 students on the study were sleep deprived.

Your Turn 3.81

Riley has a daily calorie intake of 2,200 calories and wants to take in 20% of their calories as protein. How many calories of protein should be in their daily diet?

Example 3.82: Amazon Prime Subscribers

There are 126 million users who are U.S. Amazon Prime subscribers. If there are 328.2 million residents in the United States, what percentage of U.S. residents are Amazon Prime subscribers?

Answer

We are asked to find the percentage. To do so, we divide the percentage of the total, which is 126 million, by the total, which is 328.2 million. Performing this division and rounding to three decimal places yields $\frac{126}{328.2}=0.384$. The decimal is moved to the right by two places, and a % sign is appended to the end. Doing this shows us that 38.4% of U.S. residents are Amazon Prime subscribers.

Your Turn 3.82

A small town has 450 registered voters. In the primaries, 54 voted. What percentage of registered voters in that town voted in the primaries?

Example 3.83: Finding the Percentage When the Total and Percentage of the Total Are Known

Evander plays on the basketball team at their university and 73% of the athletes at their university receive some sort of scholarship for attending. If they know 219 of the student-athletes receive some sort of scholarship, how many student-athletes are at the university?

Answer

We need to find the total number of student-athletes at Evander’s university.

Step 1: Identify what we know. We know the percentage of students who receive some sort of scholarship, 73%. We also know the number of athletes that form the part of the whole, or 219 student-athletes.

Step 2: To find the total number of student-athletes, use $\frac{100\times x}{n}$, with $x=$ and $n=$. Calculating with those values yields $\frac{100\times 219}{73}=300$.

So, there are 300 total student-athletes at Evander’s university

Your Turn 3.83

A store declares a deep discount of 40% for an item, which they say will save $30. What was the original price of the item?